Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, March 1, 2016.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, March 1, 2016

11:00 am in 345 Altgeld Hall,Tuesday, March 1, 2016

Orienting tmf with level structure

Dylan Wilson (Northwestern)

Abstract: Following Ando-Hopkins-Rezk, we build various highly structured genera for Spin and String manifolds valued in modular forms with level structure. We will try to indicate which pieces of this argument are formal, and what must be done if we wish to give orientations by other ring spectra such as TAF or conjectural cohomology theories attached to the moduli of K3 surfaces.

12:00 pm in Altgeld Hall 243,Tuesday, March 1, 2016

Some symplectic geometry in the 3-d Einstein Universe

Virginie Charette (University of Sherbrooke)

Abstract: The 3-dimensional Einstein Universe is the conformal compactification of 3-dimensional affine Minkowski space. Interestingly, it can also be interpreted as the space of Lagrangian planes in a 4-dimensional symplectic vector space. Using a suitable dictionary, we will discuss how certain facts concerning the Einstein Universe can be nicely stated in this language.

1:00 pm in 243 Altgeld Hall,Tuesday, March 1, 2016

Formation of limiting Stokes wave from non-limiting Stokes wave

Pavel Lushnikov (University of New Mexico)

Abstract: Stokes wave is the fully nonlinear periodic gravity wave parameterized by its height. Wave of greatest height has the limiting form with 120 degrees angle on the crest as found by Stokes in 1880. Assume that $z(\zeta)$ provides a conformal map of a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half-plane of $\zeta$. Then Stokes wave is fully characterized by the complex singularities in the upper complex half-plane. The only singularity in the physical sheet of Riemann surface of non-limiting wave is the square-root branch point located on the imaginary axis at $\zeta=\zeta_c$. Corresponding branch cut defines the second sheet of the Riemann surface if we cross the branch cut. We found the infinite number of square root singularities in infinite number of non-physical sheets of Riemann surface. These singularities located both symmetrically $\zeta=+-\zeta_c$ and on diagonals (with respect to vertical axis) corresponding to different non-physical sheets of Riemann surface. Increase of the height of the Stokes wave means that all these singularities simultaneously approach the real line from different sheets of Riemann surface and merge together forming 2/3 power law singularity of the limiting wave. It was conjectured (P.M. Lushnikov, ArXiv:1507.02784) that non-limiting Stokes wave $z(\zeta)$ at the leading order consists of the infinite product of nested square root singularities which form the infinite number of sheets of Riemann surface.

1:00 pm in 345 Altgeld Hall,Tuesday, March 1, 2016

Extreme amenability and amenability of automorphism groups of generic structures

Hamed Khalilian (IPM, visiting UIUC)

Abstract: In this talk, I will focus on extreme amenability and amenability of automorphism groups of some Fraisse-Hrushovski generic structures that are obtained from pre-dimension functions. By modifying the Kechris-Pestov-Todorcevic correspondence and Moore's amenability theorem, we will see that these groups do not have either of the two properties.

2:00 pm in 347 Altgeld Hall,Tuesday, March 1, 2016

Free energy in a mean field of Brownian particles.

Xia Chen (University of Tennessee at Knoxville)

Abstract: We compute the limit of the free energy of the mean field generated by the independent Brownian particles interacting through a non-negative definite function. Our main theorem is relevant to the high moment asymptotics for the parabolic Anderson models with Gaussian noise that is white in time, white or colored in space. Our approach makes a novel connection to the celebrated Donsker-Varadhan's large deviation principle for the i.i.d. random variables in infinite dimensional spaces. As an application of our main theorem, we provide a probabilistic treatment to the Hartree's theory on the asymptotics for the ground state energy of bosonic quantum system. The talk is based on the collaborate work with Tuoc Phan.

3:00 pm in 241 Altgeld Hall,Tuesday, March 1, 2016

Permutations fixing a k-set

Kevin Ford (UIUC Math)

Abstract: Let $1\le k\le n/2$. For a randomly chosen permutation $\pi \in S_n$, what is the probability that $\pi$ fixes a set of size $k$, that is, there is some subset $I\subset S_n$ of size $k$ such that $\pi$ permutes the elements of $I$? Equivalently, what is the probability that the cycle decomposition of $\pi$ contains disjoint cycles with lengths summing to $k$? When $k=1$, this is the classical derangement problem. We show that, uniformly for all $n$ and $k$, the probability lies between two constant multiples of $k^{-c}(1+\log k)^{-3/2}$, where $c=1-\frac{1+\log\log 2}{\log 2}=0.08607...$, improving on earlier bounds on Łuczak-Pyber and Diaconis-Fulman-Guralnick. The proof uses a probabilistic model for permutations, as well as ideas from number theory motivated by an analogy between the cycle decomposition of permutations and the prime factorization of integers. Some applications will be briefly discussed (invariable generation of $S_n$, permutations contained in transitive subgroups of $S_n$). Joint work with Sean Eberhard and Ben Green.

3:00 pm in 243 Altgeld Hall,Tuesday, March 1, 2016

Holomorphic Lagrangian branes, perverse sheaves and tilting sheaves

Xin Jin (Northwestern University)

Abstract: We present a correspondence from holomorphic Lagrangian branes in the cotangent bundle of a complex manifold $X$ to perverse sheaves on $X$, through the Nadler-Zaslow correspondence. This gives a way to understand tilting sheaves on $G/B$, which are important objects in representation theory, via holomorphic branes. We give a direct construction of the branes for the big tilting sheaves on $G/B$.

4:00 pm in 243 Altgeld Hall,Tuesday, March 1, 2016

The Topological Hochschild Homology of $\mathbb{Z}$

Juan Villeta-Garcia (UIUC Math)

Abstract: In the 1980's Bökstedt introduced Topological Hochschild Homology (THH), as a variant of algebraic Hochschild Homology, where tensoring over a ground ring was replaced by smashing over the sphere spectrum. Even for discrete rings, like the integers $\mathbb{Z}$, this construction provided new invariants. Bökstedt shortly thereafter calculated the THH of $\mathbb{Z}$ and $\mathbb{Z}/p\mathbb{Z}$, by exploiting heavy topological methods. Algebraically, though, a crucial tool was a spectral sequence relating classical Hochschild Homology to THH. In this talk we will introduce the spectral sequence, and sketch out a proof. We will then hint at a method of Brun to generalize to higher quotients, $\mathbb{Z}/p^n\mathbb{Z}$.